Optimal Morphs of Planar Orthogonal Drawings II
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Van Goethem and Verbeek recently showed how to morph between two planar orthogonal drawings Γ_I and Γ_O of a connected graph G while preserving planarity, orthogonality, and the complexity of the drawing during the morph. Necessarily drawings Γ_I and Γ_O must be equivalent, that is, there exists a homeomorphism of the plane that transforms Γ_I into Γ_O. Van Goethem and Verbeek use O(n) linear morphs, where n is the maximum complexity of the input drawings. However, if the graph is disconnected their method requires O(n^1.5) linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with O(n) linear morphs while preserving planarity, orthogonality, and linear complexity of the intermediate drawings. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called spirality s = O(n) of Γ_I relative to Γ_O and describe a morph from Γ_I to Γ_O using O(s) linear morphs. We prove that s+1 linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs. The resulting morphs are quite natural and visually pleasing.
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